Sequences and Series — Class 11 solved problems

34 problems from this chapter, each solved step by step.

Find the sum of an infinite geometric series: 1 + 1/3 + 1/9 + 1/27 +...
The sum of the infinite geometric series is (3)/(2).
Prove by mathematical induction that 1³ + 2³ + 3³... + n³ = [n(n+1)/2]².
By the principle of mathematical induction, 1³ + 2³ + 3³ + n³ = [(n(n+1))/(2)]² for all positive integers n.
Find the natural number a for which Σ_k=1^(n) f(a+k)=16(2^(n)-1), where the function f satisfies f(x+y)=f(x). f(y) for all natural numbers x, y and further f(1)
The natural number a is 3.
Determine the convergence of the series Σ(n=1 to ∞) (n!)²/(2n)! using the ratio test.
The series converges.
Find the sum of first n terms and the sum of first 5 terms of the geometric series 1+(2)/(3)+(4)/(9)+
The sum of the first n terms is S_n = 3(1-((2)/(3))^n) and the sum of the first 5 terms is S_5 = (211)/(81).
(i) If a, b, c, d are four distinct positive quantities in A.P., then show that b c>a d (ii) If a, b, c, d are four distinct positive quantities in G.P., then s
(i) bc > ad (ii) a+d > b+c
Evaluate the infinite product Π(n=1 to ∞) (1 - 1/n²) and prove its convergence.
The infinite product _n=2^(∞) (1 - (1)/(n²)) converges to (1)/(2).
If the sum of the first 20 terms of the series (4 1)/(4+3 1²+1^4)+(4 2)/(4+3 2²+2^4)+(4 3)/(4+3 3²+3^4)+ is (m)/(n), where m and n are coprime, then m+n is equa
421
The sum of first three terms of a G.P. is (13)/(12) and their product is -1. Find the common ratio and the terms.
The common ratios are r = -3/4 and r = -4/3. The terms of the G.P. are either 4/3, -1, 3/4 or 3/4, -1, 4/3.
If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the numbers.
The two numbers are 4 and 16.
What is the 20^ th term of the sequence defined by a_n=(n-1)(2-n)(3+n)?
The 20^ th term of the sequence is -7866.
If a_1, a_2,, a_n are in A.P. with common difference d (where d ≠ 0); then the sum of the series sin d(cosec a_1 cosec a_2+cosec a_2 cosec a_3+ +cosec a_n-1 cos
The sum of the series is cot a_1 - cot a_n.
If 7 = 5 + (1)/(7) (5 + α) + (1)/(77) (5 + 2α) + (1)/(77) (5 + 3α) + ∞, then the value of α is:
The value of α is (850)/(847).
Let a_1, a_2, a_3, be a G.P. of increasing positive terms. If a_1 a_5 = 28 and a_2 + a_4 = 29, then a_6 is equal to:
784
EXTENSION: Determine the convergence of the series Σ(n=1 to ∞) (n!)²/(2n)! using the ratio test. Additionally, generalize your solution to n dimensions.
The series Σ_n=1^(∞) ((n!)²)/((2n)!) converges. The generalized series Σ_n=1^(∞) ((n!)^k)/((kn)!) converges for k ≥ 2.
If the sum of the first 10 terms of the series (4 · 1)/(1 + 4 · 1^4) + (4 · 2)/(1 + 4 · 2^4) + (4 · 3)/(1 + 4 · 3^4) + is (m)/(n), where gcd(m, n) = 1, then m +
441
Let a_n be a sequence such that a_0 = 0, a_1 = (1)/(2) and 2 a_n+2 = 5 a_n+1 - 3 a_n, n = 0, 1, 2, 3,. Then Σ_k=1^(100) a_k is equal to
3 ((3)/(2))^(100) - 103
If f(x) = x²2² + √2, x R, then Σ_k=1^(81) f((x)/(82)) is equal to
81x²6724(4 + √2)
In a G.P., the 3^ rd term is 24 and the 6^ th term is 192.Find the 10^ th term.
The 10^(th) term is 3072.
Consider the sequence defined by a₁ = 1, aₙ₊₁ = 1 + 1/aₙ. Prove that this sequence converges and find its limit.
The sequence converges to the limit (1 + √5)/(2).
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then P² R³: S³ is equal to (A) 1: 1 (B) ( common ratio)^(n): 1 (C) (
1: 1
How many terms of the G.P. 3, (3)/(2), (3)/(4), are needed to give the sum (3069)/(512)?
10
Write the first three terms in each of the following sequences defined by the following: ll (i) a_n=2 n+5, (ii) a_n=(n-3)/(4).
The first three terms for sequence (i) are 7, 9, 11. The first three terms for sequence (ii) are -(1)/(2), -(1)/(4), 0.
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is (A) (-4)/(5) (B) (1)/(5) (C) 4 (D)
4
Let x_1, x_2, x_3, x_4 be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from x_1, x_2, x_3, x_4, then the resulting numbers are in an ar
216
Find the 10^ th and n^ th terms of the G.P. 5, 25,125,...
The 10^ th term is 9,765,625 and the n^ th term is 5^n.
Find the sum of the sequence 7, 77, 777, 7777,... to n terms.
S_n = (7)/(9)((10(10^n - 1))/(9) - n)
The minimum value of the expression 3^(x)+3^(1-x), x R, is (A) 0 (B) (1)/(3) (C) 3 (D) 2 √3
2√3
Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively, and the sum and the
540
Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.
The two possible sets of three numbers to be inserted are 4, 16, 64 or -4, 16, -64.
If a, b, c, d and p are different real numbers such that (a²+b²+c²) p²-2(a b+b c+c d) p+(b²+c²+d²) ≤ 0, then show that a, b, c and d are in G.P.
The numbers a, b, c, and d are in Geometric Progression.
In a G.P. of positive terms, if any term is equal to the sum of the next two terms. Then the common ratio of the G.P. is (A) sin 18^(°) (B) 2 cos 18^(°) (C) cos
The common ratio of the G.P. is 2 sin 18^(°).
A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.
The number of ancestors during the ten generations preceding his own is 2046.
If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of
757

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